High frequency response multilayer heat flux gauge configuration

ABSTRACT

Double-sided, high-frequency response heat flux gauge for use on metal turbine blading consists of a metal film (1500A) resistance thermometer sputtered on both sides of a thin (25 μm) polyimide sheet. The temperature difference across the polyimide is a direct measure of the heat flux at low frequencies, while a quasi-ID analysis is used to infer the high-frequency heat flux from the upper surface history.

STATEMENT OF GOVENMENT INTEREST

The invention described herein may be manufactured and used by or forthe Government of the United States of America for governmental purposeswithout the payment of any royalties thereon or therefor.

BACKGROUND OF THE INVENTION

(1) Field of the Invention

The present invention generally relates to instrumentation. Moreparticularly it comprises thin film resistance temperature sensors formeasuring heat flux to a surface at frequencies ranging from DC to overone hundred kilohertz.

(2) Description of the Prior Art

Accurate measurement of the heat transfer rate to turbine blading haslong been recognized as a key to the improvement of high pressureturbine stages. The development of heat flux instrumentation for hot,warm and cold machines has therefore been an active research area. Inhot turbines the harsh, oxidizing environment is a severely limitingconstraint. Here thermal gradient devices such as Gardon gauges arebecoming increasingly practical for measurement of the time average heatflux level.

The development of short duration turbine test facilities (30 to 700 mstest times) over the past decade has stimulated the development of heatflux instrumentation specifically optimized for the relatively benignenvironment (500°-800° K. gas temperatures) typical of these facilities.Calorimeter and thin film semi-infinite instruments have beenextensively used in these applications.

SUMMARY OF THE INVENTION

The object of the invention was the development of a heat transfer gaugetechnology which would permit simultaneous measurement of both thesteady state and time resolved heat flux distribution about the rotorblades in the MIT Blowdown Turbine facility. The specific requirementsfor this instrumentation are: that it be compatible with the rigenvironment (500° K. gas temperature, 290° K. metal temperature, 200KW/m² leading edge heat flux levels); that it be capable of withstandingthe high centrifugal stresses inherent to the rotor environment; that ithave frequency response extending from DC to 5 to 10 times blade passingfrequency: that it be usable in relatively large numbers per airfoil(10-20 per spanwise station); that it require minimal modification tothe blading; and that it introduce the minimum disruption to, orinterference with, the flow field and heat transfer.

The system described herein resulted from double-sided, high frequencyresponse heat flux gauge technology developed specifically for use onmetal turbine blading in short duration turbine test facilities. Thegauge configurations comprise a metal film (1500 A) resistancetemperature sensors sputtered on both sides of a thin (25 μm) polymidesheet. This sheet, containing 25 gauge configurations, is thenadhesively bonded to and completely covers the airfoil surface. Thetemperature difference across the polyimide is a direct measure of theheat flux at low frequencies, while a quasi-1D analysis is used to inferthe high frequency heat flux from the upper surface temperature history.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic cross-section of a multilayer heat flux gaugeaffixed to a blade surface;

FIG. 2 shows the geometry of gauge and blade surface of FIG. 1 in ananalytical model;

FIGS. 3a and 3b show the harmonic frequency response of the gaugenormalized to the direct or shunt mode cutoff frequency, ω_(c) as afunction of fractional glue thickness;

FIG. 4 shows the semi-infinite mode normalized time response of upperand lower sensors to a step in heat flux as a function of fractionalglue thickness;

FIG. 5 shows the direct mode normalized response to a step input of heatflux as a function of fractional glue thickness;

FIGS. 6a-e show the gauge geometry including sensor detail, sensor andleads of a polyimide substrate;.

FIG. 7 shows the gauge thermal calibration data, √ρck, in proportionalto the difference in slope of the curves;

FIG. 8 shows the rise time of the numerical data reduction model as afunction of model bandwidth;

FIGS. 9a and 9b shows the frequency response of 100 kHz bandwidth datareduction numerical scheme;

FIGS. 10a-f show the data reduction scheme response as a function of thenumber of calculation nodes;

FIG. 11 shows the time history of gauge sensors mounted on thestationary casing above a transonic turbine rotor tip; and

FIG. 12 shows the heat flux calculated from the data in FIG. 11 startingat 300 ms.

DESCRIPTION OF THE PREFERRED EMBODIMENT

Four candidate techniques were considered in some detail; calorimeter,blade temperature conduction models, thin film semi-infinite, andmultilayer thin film gauges. All four techniques have been demonstrated,at least in stationary measurements, and are probably realizable in therotating frame.

The technique selected was that of the multilayer thin film gauges.Referring now to FIG. 1 there is shown a single multilayer heat fluxgauge configuration that is comprised of two temperature sensors 12 and14 bonded on either side of a thin insulating substrate 16. A pluralityof the gauge configuration 10 are affixed to a blade surface 18 by meansof any suitable adhesive 20. The configuration 10 provides a thermalshunt. The temperature difference across the insulating substrate 16 isa direct measure of the heat flux to the blade surface 18 below acertain frequency. This is due to the direct proportionality between thetemperature difference as measured by sensors 12 and 14, and the heatflux being valid only below this certain frequency. This frequency,however, increases as the insulating substrate 16 thickness is reduced.Conversely, above another frequency, the substrate 16 appearssemi-infinite to the upper sensor 12 and a quasi-one-dimensionalassumption can be used to infer the heat flux. In fact, the multilayergauge configuration 10 could be considered an elaboration of thesemi-infinite gauge in which a second sensor is placed within thesubstrate to eliminate the gauge performance dependence on the bladematerial properties. Alternatively, the gauge configurations 10 can beviewed as developments of the commercially available thermopile sensor,low frequency response gauges with the low response sensors replaced byhigh response resistance temperature sensors 12 and 14. For theselection of 25 μm thick polyimide insulator for substrate 16, thedirect response (shunt) mode frequency response is DC to 20 Hz, whilethe semi-infinite assumption is good above 1.5 kHz. In the MIT BlowdownTurbine tunnel for which these gauge configurations 10 were designed,rotor blade passing frequency is 6 kHz. As later explained, the responsein the intermediate 20-1500 Hz region can be reconstructed throughproper numerical signal processing.

The multilayer gauge configurations 10 are fabricated many to a sheet ofpolyimide insulating material 16, tested, and then glued to the blade ortest article surface 18 using standard strain gauge adhesive bondingtechniques. The blade 18 is completely covered by the polyimide sheetmaterial 16. The specific advantages of this heat flux gauge technologyinclude: frequency response from DC to tens of kilohertz, conventionalmetal blading may be used, no thermal or geometric discontinuities areintroduced, and the gauges 10 are fabricated and tested independently tothe airfoil.

This disclosure details the theory, fabrication, testing and datareduction of these high frequency response multilayer heat flux gaugeconfigurations 10.

The construction of a generic multilayer, time dependent, conductionmodel of the double sided heat flux gauge configuration 10 to use in thegauge design and calibration, and as a guide to its proper applicationis now explained. A cross-section of a mounted gauge configuration 10 isillustrated in FIG. 1. When mounted on a blade profile 18 or test item,the gauge configuration 10 becomes part of a five-layered structureconsisting of the top thin film thermometer temperature sensor 12, thegauge insulating substrate 16, a second film temperature sensor 14, anadhesive layer 20, and finally the test article 18 itself. In general,the thermal properties of each layer will differ and thus, to be fullyrigorous, a five-layer model might be employed. In practice, however,the film thermometer sensors 12 and 14 are sufficiently thin to appearthermally transparent to the applied heat flux, thereby permitting useof a three-layer model. Also, the sensors 12 and 14 will be excited atpower dissipation levels sufficiently low that their influence as sheetsources of heat can be neglected. The quantitative conditions requiredto meet these criteria will be derived later.

The model can be further simplified to two layers, as shown in FIG. 2 ifwe assume that the sensor substrate 16 and adhesive 20 have similarthermal properties and that the sensors are embedded in the upper layer1ρ₁, c₁, k₁. In detail, this adhesive thickness will represent not theexact physical thickness but an equivalent thickness including effectsof uneveness and property variations. A similar assumption will be madeabout any surface coating 24 (protective or dirt) which may be on top ofthe gauge configuration 10 surface. The adhesive 20, substrate 16 andsurface coating 24 will therefore form upper layer 1ρ₁, c₁, k₁ of themodel and the test article or blade 18 will form lower layer 2ρ₂, c₂,k₂. Layer 1ρ₁, c₁, k₁ is subsequently referred to as layer 1 and layer2ρ₂, c₂, k₂ is subsequently referred to as layer 2. The upper sensorT_(u) 12 will lie under a surface layer 24 of thickness h=f_(h) d andthe lower sensor T_(l) 14 will lie a distance g=f_(g) d, representingthe glue layer 20, above the interface 25, where d represents theseparation of sensors 12 and 14 and f_(g), f_(h) represent the glue 20and surface 24 coating fractions respectively. Thus x_(u) =f_(h) d andx_(l) =(l+f_(h))d, and if L is the thickness of upper layer 1, then(l+f_(h) +f_(g))d=L.

The heat conduction equations will be solved for the temperaturedistribution in a two layered semi-infinite medium of unlimited lateralextent subject to a spatially uniform surface heat flux, q(t). The flowof heat may therefore be considered to be one-dimensional. It is furtherassumed that the thermal properties within each layer 1 and 2 areuniform and that the layers 1 and 2 make perfect thermal contact. Thus,the temperature in each layer 1 and 2 is governed by, ##EQU1## (whereκ=k/ρc is the thermal diffusivity and k=thermal conductivity, ρ=densityand c=specific heat, subject to the interface conditions, ##EQU2## andthe semi-infinite condition,

    T.sub.2 (∞,t)=0.                                     Eq. (5)

To fully specify the problem, upper surface boundary conditions andinital conditions must be provided and these will depend upon the formof the driving surface heat flux to be investigated. We will considertwo cases: (1) a steady sinusoid and (2) a step in surface heat flux.The solution to these problems employs standard techniques. The resultsare given in a form useful to the subsequent analysis and only thesolution for the temperature in the layer containing the sensor, T₁(x,t), will be reported.

In response to a steady harmonic variation in surface heat flux theinitial conditions cannot be specified since the steady solution validfor all time is being sought. for the surface heat flux, ##EQU3## thetemperature distribution in Layer 1 is,

    T.sub.1 (x,t)=(Q.sub.o d/k.sub.1)M exp {j(ωt-φ)}0≦×≦L              Eq. (7)

where, ##EQU4## A sensor characteristic frequency ω_(c) has beenintroduced based upon the sensor spacing d. ##EQU5## The otherquantities are defined as follows: ##EQU6##

In response to a step in surface heat flux both the transient and steadysolution to the problem are of interest. The initial conditions are,##EQU7## For the surface heat flux, ##EQU8## the temperaturedistribution in Layer 1 is given by, ##EQU9## and ierfc() is the firstintegral of the complementary error function. A gauge time constant γhas been introduced, again based upon gauge thermal diffusivity andsensor spacing.

    τ≡d.sup.2 /4κ.                             Eq. (18)

The temperature is referenced to the quantity (Q_(o) d/k₁) whichrepresents the steady state temperature drop required to "drive" a heatflux Q_(o) across the distance d.

The characteristic frequency ω_(c) and the time constant γ are definedwith respect to the gauge film spacing d and not the layer thickness L.Thus, the expressions for the solution may appear a little morecumbersome than necessary to describe the temperature distribution butthey are in the proper form for the subsequent analyses of the gaugeresponse characteristics.

In discussion of the gauge model the solutions for the temperature inLayer 1, T₁ (x,t), depend intrinsically upon the properties of the gaugethrough √ρck and k/d. These parameters form the basis for thenormalization of the independent variables through the definitions of γand ω_(c). We will adopt these as the fundamental parameters whichcharacterize the gauge and which, therefore, must be known to interpretits output. Additional influence comes from the thickness fractions ofthe surface coating and adhesive (f_(h) and f_(g) respectively) and the√ρck of Layer 2 which enters through the parameter R.

The gauge model is now used to show that only the film thermometer scalefactor, √ρck and k/d need calibration. Variations in adhesive thicknessand test article properties (f_(h), f_(g), R), although they influencethe temperature levels, do not significantly affect the heat flux levelinferred.

For gauge model response there is used the limiting case solutions ofthe low frequency direct mode and the high frequency semi-infinite mode.This provides a framework for gauge design and calibration.

In the direct or shunt mode, the surface heat flux is assumed to beproportional to the measured temperature drop across the gaugeconfiguration 10. While this is valid for static measurements, a uniformtemperature gradient will not exist across the gauge configuration 10 ifthere are frequency components for which the thermal penetration depthis comparable to, or smaller than, the gauge configuration 10 thickness.The gauge configuration 10 is considered to be in the semi-infinite modefor those frequency components which are effetively damped by the gaugesubstrate 16. In this case, the heat flux may be obtained entirely fromthe upper film thermometer T_(u), 12. It should be understood that`direct` and `semi-infinite` are terms used to describe limitingprocesses within the gauge substrate 16 and are in no way exclusive ofeach other in actual operation of the gauge configuration 10.

For steady harmonic excitation in the direct measurement mode, the upperand lower film temperatures are provided by Eq. (7),

    T.sub.u =T.sub.1 (x.sub.u,t)=(Q.sub.o d/k.sub.1)M.sub.u e.sup.j(ωt-φ.sbsp.u.sup.)                       Eq. (19a)

    T.sub.l =T.sub.1 (x.sub.l,t)=(Q.sub.o d/k.sub.1)M.sub.l e.sup.j(ωt-φ.sbsp.l.sup.)                       Eq. (19b)

where M_(u) =M(x_(u)), etc.: x_(u) =f_(h) d and x_(l) =(l+f_(h))d arethe positions of the upper and lower temperature sensors 12 and 14; andL=(l+f_(H) +f_(g))d is the total thickness of Layer 1. The direct modeheat flux q_(D) indicated by an `ideal` gauge, (i.e., error-freemeasurements of T_(u) and T_(l), and perfect knowledge of k₁ /d), is##EQU10## Substituting for the temperatures using Eqs. (19), the ratioof the measured to actual heat flux is, ##EQU11## This may be put intothe more useful magnitude-phase form, ##EQU12## where

    M.sub.D ≡{M.sub.u.sup.2+M.sub.l.sup.2 -2M.sub.u M.sub.l cos(φ.sub.u -φ.sub.l)}.sup.1/2                    Eq. (22)

    φ.sub.D ≡tan.sup.-1 {[M.sub.u sin φ.sub.u -M.sub.l sin φ.sub.l ]/[M.sub.u cos φ.sub.u -M.sub.l cos φ.sub.l ]}Eq. (23)

represent the magnitude M_(D) and the phase φ_(D) response for directmode.

These are plotted in FIG. 3 as a function of the normalized frequencyω/ω_(c), for several values of glue fraction, f_(g). The case shown isfor a polyimide substrate mounted upon an aluminum test body, R=0.96.The results show that the direct temperature difference mode is valid,in this example, for frequencies up to the cutoff frequency ω_(c), andthat in this region the adhesive layer has negligible influence upon theperformance of the gauge. Parametric studies show similar conclusionsfor the influence of R and f_(h).

The results thus far show that, in the direct mode, the temperature riseper unit of surface heat flux can be approximated by, ##EQU13## up tothe cutoff frequency, ##EQU14## If k/d is eliminated from theseexpressions, a constraint between gauge sensitivity ΔT/Q and theresponse bandwidth f_(c) (in Hz) is obtained, ##EQU15## i.e., theoverall trade between signal strength and cutoff frequency depends onlyon the thermal √ρck of the gauge substrate. Thus, both low thermalinertia ρc (more precisely high diffusivity), good for high f_(c), andlow thermal conductivity, good for high ΔT/Q, are desirable.

The solution for steady harmonic excitation, Eq. (19b), shows that theresponse of the lower sensor is fully damped for frequencies aboveapproximately 100 ω_(c). In the case of the semi-infinite mode, theexpression for the surface temperature, Eq. (19a) (for x=0) reduces to##EQU16## Thus, the condition for maximum high frequency temperaturesensitivity is the same as for the direct measurement mode, low √ρck,except now it is seen that sensitivity also diminishes with frequency,as ω^(-1/2). This means that, for a fixed temperature measurementcapability, low values of √ρck will help to improve the upper frequencyresponse limit of the semi-infinite mode.

In summary, this steady sinusoidal theory provides the basis for thegauge design, i.e., substrate and thickness selection. Low √ρck isdesirable for a high sensitivity-bandwidth product in direct measurementmode and for high sensitivity and frequency response in thesemi-infinite mode. For a given √ρck, the trade between f_(c) and ΔT/Qis then set by the choice of substrate thickness, d.

Now a look is taken at the semi-infinite mode step response, which isshown for the upper and lower thermometers in FIG. 4. The region inwhich the response of the upper thermometer is linearly proportional to√t is that for which the semi-infinite assumption is valid. Here, thesolution for x=0 is, ##EQU17##

This equation is used for √ρck calibration procedures discussed later.Note that, although increasing the adhesive fraction tends to extend theduration of the semi-infinite regime, it does not influence theperformance of the gauge while in that regime. similar conclusionsfollow for R and f_(h).

The step response for the direct difference mode is obtained from Eqs.(16) and (20) (FIG. 5). The effect of the glue layer is more evidentfrom its influence upon the rise time of the measured response than itwas from its effect upon the frequency response. Also, shown is adiscriminant t_(u) -T_(l) /T_(u), which can be used to estimate theeffective thickness of the adhesive from test data.

These results clearly show the behavior of the semi-infinite (t≦ρ) andsteady state (t>20τ) limits. The actual data is reduced with thenumerical technique which reconstructs the entire frequency domain, aswill be described later.

This section discusses the design of thin film resistance thermometersused as temperature sensors 12 and 14 in the multilayer heat flux gaugeconfiguration 10. These sensors 12 and 14 are nothing more than metalfilm resistors whose resistance changes with temperature. For a changeδT about a temperature T, the increment in film resistance may beexpressed by a Taylor series expansion in power of δT, ##EQU18## Thetemperature coefficient of resistivity α(T), a fundamental materialproperty, represents the fractional change in resistance about a giventemperature,

    α(T)=d(ln R)/dT.                                     Eq. (31)

For the metals commonly employed in resistance thermometers, α(T) isusually a very weak function of temperature over ranges of practicalinterest. Thus, in many applications, a convenient form of Eq. (30) maybe obtained by assuming constant α and neglecting the higher orderterms,

    ΔR=RαδT.                                 Eq. (32)

(For the sensor here, this results in errors of order 0.1C over a 60Crange.)

For a sensor excited at constant current, the change in voltage acrossthe sensor, δV, is

    δV=(Vα)δT,                               Eq. (33)

where V represents the total voltage drop across the film. Thus thesensitivity of the sensor is directly proportional to both itstemperature coefficient and the excitation voltage. The excitationvoltage is constrained, however, by the V² /R heat dissipation in thethe film sensor,

    Q.sub.E =V.sup.2 /(Rlw),                                   Eq. (34)

(where l and w are the active length and width of the film). This heatdissipation must be kept small compared to the heat flux being measured.

To quantify this constraint, film resistance will be expressed in termsof film geometry and volume resistivity,

    R=(ρl)/(wt),                                           Eq. (35)

where t is the film thickness. Substituting into Eq. (34) yields

    V=(ρ/t).sup.1/2 lQ.sub.E.sup.1/2.                      Eq. (36)

This expression sets the excitation voltage. Note that this result isindependent of the width of the film. The level of Q_(E) can be adjustedfor each measurement application. The film parameters (ρ/t, l, w, t),however, must be selected beforehand by the design process. Substitutionof Eq. (36) into Eq. (33) yields a form of the sensitivity equationuseful for design,

    δV/δT=(ρ/t).sup.1/2 αlQ.sub.E.sup.1/2. Eq. (37)

This expression implies that long, thin films with high volumeresistivity and temperature coefficient are best for high temperaturemeasurement sensitivity. There are, however, many other conditions whichmust be considered to arrive at an overall optimum design of filmresistance sensors for heat transfer gauges, some of which are discussedbelow.

The first two factors in Eq. (37), (ρ/t)^(1/2), show the influence offilm thickness and electrical properties on film sensitivity. Candidatefilm materials must also be raked based upon the sensitivity parameterα√ρ, derived using the bulk properties of the materials. Note, however,that these properties are a function of film thickness in the regionbelow 1 μm where data is available for only a few substances. Availabletables do provide some general guidance for maximizing temperaturesensitivity, the final choice being also influenced by:

(a) surface temperature changes resulting from the thermal resistanceand capacity of the film,

(b) abrasion resistance and substrate adhesion of the film,

(c) residual stresses induced by the deposition process, and

(d) changes in ρ and α from their known bulk property values.

The third factor in Eq. (37) shows the influence of sensor surfacegeometry. The maximum length of the sensor, hence its sensitivity, willbe determined by the spatial resolution requirements of the heat fluxmeasurement. If this is characterized by a single length scale λ, forexample, the film will be constrained to lie in a roughly square region,λ on a side. Since the sensitivity has been shown to be independent offilm width, a serpentine pattern can be employed to significantlyincrease the film sensitivity, if the minimum width that can be reliablydeposited is much smaller than λ. For the film geometry chosen, l≈14λ.There are, however, several potential disadvantages to those thin, longand narrow film sensors. First, they are more vulnerable to small scalenicks and scratches. Second, they may prove unsuitable for calibrationof substrate physical properties by the electrical self-heating method,which requires that the film width be much larger than the thermalpenetration depth over the required calibration time. Third, thesensitivity advantage of a long, thin, narrow film can be reduced by thereduction in its inherent signal to noise ratio which arises due to itsincreased resistance.

The following is an estimation of the signal-to-noise ratio of this thinfilm resistance thermometer. All resistors produce electrical noise as aresult of the thermal motion of their electrons. The mean squared valueof this thermal noise voltage is V_(t) ² =4kTBR; where k is theBoltzmann constant, T the resistor absolute temperature, B the bandwidthover which this (white) noise voltage is measured, and R is theresistance. A second contributor to the resistor noise arises when theresistor `feels` the flow of an externally applied current. Allelectrical currents have inherent fluctuations, arising from the factthat change comes in discrete lumps (electrons). For a current of(average) magnitude I, the mean squared value of these fluctuations isi_(s) ² =2eIB, where e is the electron charge. Passing through aresistor, these fluctuations appear as a voltage noise V_(s) ² =i_(s) ²R² =2eVRB, shot noise. Since these sources are uncorrelated, the totalresistor noise may be expressed as,

    V.sub.n.sup.2 =V.sub.t.sup.2 +V.sub.s.sup.2 =(4kT+2eV)RB.  Eq. (38)

Note that V_(n) ², a measure of the noise power, is proportional to theproduct of the observation bandwidth with the resistance (and implicitlywith the gain of the film sensor).

The mean squared signal power (obtained from Eq. (37)) is,

    δV.sup.2 =(ρ/t)α.sup.2 l.sup.2 Q.sub.E δT.sup.2 Eq. ( 39)

Forming the ratio of mean squared signal-to-noise powers yields thesignal-to-noise ratio, ##EQU19## where Eqs. (35) and (36) have been usedfor R and V in Eq. (38)

The film noise will result in an indicated RMS temperature fluctuation,δT_(n) which may be found from Eq. (40) by setting S/N=1, ##EQU20## Forthe nominal operating conditions of the sensors designed here, thistemperature is equal to 0.002° K., quite negligible. A more rigorousoptimization of film design would include all the tradeoffs between thefilm, amplifier, and data acquisition system. This was not done in thiscase since this gauge noise level is less than the 0.08° K. peak-to-peakequivalent noise of the amplification system used.

Refer now to FIGS. 6a-e for the fabrication and mounting of gaugeconfigurations 10. Given the guidelines implicit in the gauge theory,the gauge configuration 10 design consisted of selection of thesubstrate 16 material, the serpentine pattern film resistor 30 material,and the sensor 12 and 14 geometry. All three must be compatible with thefabrication technique selected. Vapor deposition and photolithographywere chosen as being compatible with both large scale, low costfabrication of these gauges 10 and the fine sensor structure implied bythe relatively high film sensor resistance desired.

Over fifty materials were studied as candidates for the gauge substrate16. Polyimide (Kapton) was by far the most attractive material. It hasthe lowest √ρck of any material studied (and thus the highest sensitvitybandwidth product), has excellent thermal and mechanical properties upto 600° K., has a well-developed adhesive technology (due to its wideuse in strain gauges), is widely available, and is used in vapordeposition applications for spacecraft. The substrate 16 thickness, d,was selected as 25 μm to yield a direct mode response upper limit of 20Hz and a semi-infinite mode lower limit of 1500 Hz.

The resistance temperature sensor serpentine patterns 30 are fabricatedfrom 0.13 μm thick pure nickel. Nickel was chosen because of itsrelatively high net sensitivity and because of its very good adhesioncharacteristic in thin film applications. Platinum and titanium werealso found to be suitable materials. The sensor 12 geometry selected wasa square serpentine pattern to maximize the sensor length in an areacompatible with the expected disturbance length in the flow field andthus maximize the signal-to-noise ratio as discussed earlier. Theserpentine pattern results in a negligible heat flux when the resistancetemperature sensor serpentine patterns 30 are measured. The sensorresistance is approximately 500Ω. Low resistance gold leads or `tags` 32that are 1 μm thick are deposited from the sensor to the edge of thepolyimide sheet material 16. Signals from the bottom sensor 14 comethrough 0.5 mm diameter `plated through` holes 34, laser drilled in thesubstrate 16 before the deposition process. Twenty-six gaugeconfigurations 10 are fabricated on a single polyimide sheet material 16whose dimensions are compatible with a turbine airfoil surface area.

The vapor deposition is done by DC sputtering at a pressure of 5×10⁻⁴torr of argon. The deposition rate of the nickel is 0.2 nm per second.The geometrical delineation is done using a liftoff process, the twogauge configuration 10 sides being done separately. The manufacturingyield and material properties are very strongly process variabledependent.

Originally, a silicon dioxide overcoat was applied to the bottom sensor14 surface to serve as an electrical insulator. This was discontinued infavor of anodizing the surface of the aluminum blading, a practice whichhas proven completely satisfactory.

The polyimide material 16 are simply cut to conform to the blade surfaceoutline and then bonded to the surface using conventional strain gaugecement and mounting techniques. Thermal time response testing hasindicated that the glue layer is 5 μm thick. Seventy-five microndiameter wires (not shown) are soldered to the far ends of the gold tagsto bring out the signals through slip rings (not shown) in a region farremoved from the measurement area.

The following is a description of the calibration of the heat fluxgauges. As in most experimental techniques used to determine heat flux,temperature is the physical quantity measured, from which surface heatflux is inferred by one of the several techniques: direct processing ofthe temperature signal by a physical RC analog network, numericalintegration of the measurement with a kernal function, or processing ofthe measurement of a numerical analog of the gauge configuration 10 heatflow. These techniques are based upon a model of the heat conductionprocess within both the gauge configuration 10 and, in most cases, thetest article. The model not only provides the physical and mathematicalframework for each approach but also the form of the fundamentalparameters which govern the behavior of the gauge.

The multilayer gauge configuration 10 provides two temperaturemeasurements: at the surface, and at the depth d below the surface. Fromthe model shown in FIG. 2 derived earlier, it can be seen that twoconstants must be determined -√ρck (the conventional parameter used withsurface thermometry based upon the semi-infinite assumption), and k/d(representing the steady state heat flux per unit temperature dropacross the gauge configuration 10).

Careful calibration of these parameters is particularly important forthis gauge since there is far less experience with polyimide as asubstrate than with the more conventional quartz, Pyrex, or machinableceramic materials. Significant deviations from the `nominal` oradvertised thermal and physical properties might occur within themanufacturing tolerances of the polyimide sheet 16. Furthermore, it isnot yet known what influence, if any, the film thermometer depositionprocess might have upon these properties. Thus, given that the upper andlower film sensors 12 and 14 have been calibrated as thermometers andthat all the assumptions pertaining to the sensor model are satisfied,knowledge of √ρck and (k/d) will fully characterize the operation of theheat flux sensor 10.

The temperature coefficient of resistivity of the thin film temperaturesensors 12 and 14 is conventionally calibrated by placing the gaugeconfigurations 10 in a heated immersion bath and varying the bathtemperature. All sensors on a test specimen are tested together. Theaccuracy of the calibration is limited by that of the referencethermometer to approximately 0.05° C.

Many calibration schemes for √ρck and k/d place boundary conditions onthe gauge configuration 10 for which exact solutions of the heatconduction equation can be found. The calibration coefficients may thenbe inferred from a simple comparison of the experimental data with thetheoretical solution. The overall accuracy of the result will bedependent upon both the quality of the various physical measurementsrequired and the accuracy to which the assumed boundary conditions arerealized. This latter condition is especially crucial since its validitymay not necessarily be apparent from the test data alone. Thus, there isa practical virtue in making these boundary conditions as simple aspossible, i.e., that the flow of heat be one-dimensional and that thegauge 10 appear thermally semi-infinite. Given these conditions, simpleanalytical solutions can be found for the surface temperature responseto a wide variety of surface hear flux time histories. For example, thetop surface response to a step in surface heat flux, Q, is given by,##EQU21##

Commonly, the step in surface heat flux is applied by pulsing currentthrough the film, i.e., resistive dissipation in the sensor itself. Thistechnique cannot be used with this gauge 10 geometry, however, since thetemperature sensor 12 or 14 width is on the order of the insulatorthickness, violating the uniform heating assumption and introducingerrors on the order of 100%. Instead, a calibration technique is used inwhich the gauge 10 is radiantly heated by a laser pulse, simultaneouslyyielding √ρck and k/d.

Conceptually, step radiant heating can be used as a direct calibration,i.e., a known heat rate is applied to the top sensor 12 with √ρck beinginferred from the top sensor 12 rise time and k/d simply computed fromthe top 12 and bottom 14 sensor temperatures. In practice, this can bedifficult to achieve since the measurement accuracy is directlyproportional to the absolute accuracy to which the pulse power is known,dependent upon absorption characteristics of the surface, and issensitive to the energy distribution in the laser beam. Rather thanattempt to perform an accurate absolute measurement, a relativecalibration technique was developed which depends upon ratios, notabsolute values.

In the relative calibration, the step response is measure with thesensor both covered and uncovered by a reference fluid of known √ρck.The method is based upon the principle that the applied surface coatingacts as a sheet source of heat at the interface of two semi-infinitesubstances. Theory shows that the ratio of heat entering each substanceequals the ratio of their respective values of √ρck (since bothmaterials see the same interface temperature), and this fact may beexploited to measure this ratio. Since the method involves comparison ofmeasurements, absolute knowledge of the incident heat flux and filmthermometer scale factor are not needed, being replaced instead onlywith the requirement that these quantities remain stable over theduration of the tests. However, any change in the total heat fluxabsorbed by the surface coating as a result of the application of thereference fluid (by mechanisms such as meniscus focusing, absorption bythe fluid or by reflections from its surface) must either be negligibleor quantifiable.

For the first set of tests, the sensor is placed in vacuum or still air.If Q_(I) represents the surface heating, the resulting temperature riseaccording to the theory is, ##EQU22##

For the second set of tests, the sensor is placed in good thermalcontact with a material of known √ρck by covering it with a fluid ofhigh electrical resistivity. If Q_(S) represents the heat flux enteringthe sensor, and Q_(F) that entering the fluid, than at their commonboundary the temperature rise is given by, ##EQU23## from which it canbe concluded that, ##EQU24## If the total heat flux absorbed by thecoating for this second series of tests is Q_(II), where Q_(II) =Q_(S)+Q_(F), the flux into the sensor is found to be, ##EQU25## and thetemperature rise at the surface is therefore, ##EQU26## If m_(I) andm_(II) are the slopes of the linear region of sensor temperature, T,versus the square root of time (i.e., from Eqs. (43) and (47)) for thetwo conditions, then ##EQU27## and by forming their ratio, ##EQU28## Ifthe total heat absorbed by the coating is the same in both tests (as hasbeen verified in this case), this relation becomes. ##EQU29## and servesas the basic relative calibration formula. Note that only ratios of thequantities appear.

A simultaneous calibration of k/d may be obtained by extending theheating time to values very much larger than the characteristic time ofthe sensor, γ=d² /4.sup.κ. The lower sensor begins to respond to thesurface heating at approximately time γ(γ=1.6 ms for these gauges 10),with steady state conditions being achieved by t=20γ. For t>20γ, thesteady state temperature difference, (T_(u) -T_(l))_(ss), between theupper and lower surfaces of the sensor becomes proportional to theapplied heat flux, thus k/d may be found from, ##EQU30## Given that Q isknown.

The magnitude of Q could be obtained from an independent measurement ofbeam intensity and surface absorptivity (as would be required for anabsolute calibration). However, when the k/d calibration is merged withthe relative procedure, Q may be calculated directly by combining theinitial step response data from the first set of tests, Eq. (48), withthe value of √(ρck)_(s) determined from the reference fluid tests, Eq.(51). Substituting this result into Eq. (52) then yields the desiredexpression for k/d), ##EQU31## It should be pointed out that, unlike the√ρck testing, this expression requires that the film thermometertemperature coefficients of resistivity be calibrated or, moreprecisely, that their scale factors, if unknown, at least be equal.

An argon ion laser was used as the radiant heat source, providing anincident flux of approximately 30 kW/m² over an area slightly largerthan the 1 mm square of the film thermometer. The laser output wasmodulated by an electro-optic modulator producing a light step with a 25ns rise time. The surface of the sensor was blackened with a StaedtlerLumograph Model-316 non-permanent marking pen, commonly used forview-graph presentations. The coating has good opacity and low thermalinertia, as verified by comparing the measured temperature responseswith the parametric theory presented earlier, and its selection was theresult of a large number of empirical tests. This coating has theadvantage, unlike many of the other coating materials evaluated, ofpossessing both high electrical resistivity and poor solubility in thereference fluids.

The reference fluid chosen was dibutylphthalate, a commonly used heatingbath medium with electrical equipment, because of its inert propertiesand high electrical resistivity. Its thermal properties were obtainedfrom the manufacturer and checked against the properties of the morecommonly employed glycerol using the electrical heating technique on aplatinum-quartz film sensor. Its value of √ρck is estimated to be 495±5%. Since it is much less viscous than glycerol, it does not producemeasurable beam focusing effects, as did the latter fluid.

A sample calibration is shown in FIG. 7, from which it can be seen thatthe top sensor 12 is linear with the square root of time. The slopes mare then calculated from the least square fits to the data asillustrated by the lines. From these measurements, we conclude that √ρckfor this gauge 10 is 575(W/m²)(√sec)/°K. and k/d is 8086 W/(M² °K). Thisrepresents a 20% variations from nominal published data.

Since the gauge configurations 10 are intended for a highly stressedrotor environment, gauge sensitivity to strain must be considered. Thiswas evaluated by pulling an aluminum specimen bearing a gauge sensor ina tensile test machine. Strain sensitivity proved to be negligible,equivalent to 0.003° C. at 2% strain.

The heat flux gauge configurations 10 have been shown to be suitable forevaluation of the surface flux to turbine blading at low and highfrequencies when the gauge characteristics can be simply defined. At lowfrequencies, the heat flux can be evaluated directly from the measuredtemperature difference across the insulator, and at high frequenciesthrough the use of semi-infinite procedures. The numerical techniquedescribed here of reducing the two temperature histories to heat fluxexpands the frequency range of the gauges to include the regime betweenDC and semi-infinite response.

As with most numerical techniques, discretization of the governingequations is required, including distribution of calculational stationsfor nodes through the insulator. In order to provide accurate predictionof the surface heat transfer rate, the distribution of the calculationnodes must be such as to accurately capture the varying temperaturefield in the insulator.

The governing heat conduction equation is: ##EQU32##

For low frequency heat transfer rate variations, the time derivative inthe above equation approaches zero, and thus a constant temperaturegradient is expected acros the insulator. Since this is a linearsolution, the exact placement of nodes in the substrate is not crucial.

As the excitation frequency increases, however, the upper surfacetemperature signal is attenuated through the insulator until, at veryhigh frequencies, the insulator appears semi-infinite. The expectedtemperature solution will be: ##EQU33## Thus, the magnitude of T decaysexponentially through the insulator and a logarithmic spacing can beused to capture the temperature profile efficiently in the numericalscheme. The numerical technique utilizes a lumped network of discreteelements of logarithmically varying thermal conductivity and thermalcapacity to calculate the surface heat transfer rates. The increasingthermal impedance of each element of the network corresponds to placingthe temperature nodes at logarithmically increasing distance into thesubstrate.

The heat conduction equation is descretized in finite-difference form(for non-uniform node spacing) as: ##EQU34## where the node number iranges from 1 to the number of sections N.

The solution of this lumped network equation is an extension of the workpublished by Oldfield et al for the design of high frequency lumped RCnetworks for the analog evaluation of heat transfer rates tosemi-infinite substrates. Oldfield et al work was concerned with thesimulation of the semi-infinite heat conduction process, thus is notdirectly applicable to the finite thickness heat flux gauges describedin this paper. The finite thickness heat flux gauge was accounted for byadding an extra thermal resistance at the end of the lumped network,thereby transforming it from a semi-infinite to a finite length line.

Equation (57) can be written in matrix form (simulating a finite lengthline) as: ##EQU35##

The finite length line now requires two inputs, the temperature of theupper surface, T_(u), of the insulator and the lower surface, T_(l). Theresulting set of coupled equations is solved by a fourth orderRunge-Kutta method to generate the upper surface heat flux rate from thecalculated temperature drop across the first element of the network.

The selection of the thermal impedance of each of the elements is set tosatisfy the following conditions:

(1) the finite thickness of the insulator must be mirrored in the totalthermal impedance of the network;

(2) ρck for each element must equal the physical value;

(3) the thermal impedance of the first element is set by the requirednetwork bandwidth; and

(4) the number of nodes required in the insulator sets the logarithmicspacing of all but the first element.

The accuracy of such a lumped network simulation can be established byback-calculating the incident heat transfer rate for exact analyticalupper and lower temperature solutions in specific cases. Two cases werestudied: a step change in heat flux, and a sinusoidal heat fluxvariation.

FIG. 8 shows the normalized heat transfer rate variation calculatedusing exact upper and lower surface temperature solutions for a stepchange in heat transfer rate applied to a gauge glued to an aluminumsubstrate. The calculated response is shown for five network upperfrequency limits, all with nine stages used to discretize thetemperature field.

FIG. 9 shows the frequency response of a 100 kHz bandwidth network withnine stages. The input temperatures were exact solutions to the heatconduction equation for a sinusoidal heat flux variation on a gaugeconfiguration glued to an aluminum substrate. The magnitude and phasedata presented in this figure is therefore relative to the driving heatflux rate variation. The agreement between the exact and numericalmethods is within 0.3% over the 5 to 2000 Hz band which is difficult torecover otherwise.

The effect of changing the number of elements used in the network on theerror in reconstructing a sinusoidal heat transfer rate for a nominal100 kHz bandwidth is whown in FIG. 10. It can be seen that the errorsare relatively insensitive to the number of stages chosen for thenetwork, and are more closely tied in with the specification of thethermal impedance of the first element in the network.

As an initial test of the gauge technology, the heat flux gauges weremounted on the stationary outer tip casing above the rotor of atransonic turbine in the MIT Blowdown Turbine Facility. A time historyof the top and bottom temperature sensors of a gauge is shown in FIG.11. At time equal to zero, the tunnel is in vacuum and the sensors arethe same temperature. After the starting transient, the tunnel operationis then quasi-steady (from 250 ms). The metal substrate temperatureremains constant while the inlet temperature slowly drops, reflectingthe isentropic expansion from the supply tank. This is seen in thedecrease in the top-bottom temperature difference and thus in heat fluxover the test time. The thick line of the top sensor output is theenvelope of the high frequency heat flux components.

The heat flux, as calculated by the numerical data reduction techniquefrom these two signals starting at 300 ms, as shown in FIG. 12. Theblade passing frequency is approximately 6 kHz and the samplingfrequency is 200 kHz. Note that the relatively small amplitude of thehigh frequency temperature fluctuation on the top sensor in FIG. 11actually represents an AC heat flux modulation of 80%. The small ACtemperature signal relative to the DC reflects the ω^(1/2) rolloff insensitivity discussed in the gauge theory.

The heat flux time variation on the tip casing was quite periodic withblade passing. This facilitated construction of a contour plot of thecasing heat flux distribution from a line of gauges arranged over theblade chord. Overall, a very high level of both means and time varyingheat flux were seen with the peak levels on the order of that at nozzleguide vane leading edge.

A multilayer heat flux gauge configuration 10 has been developed whichsuccessfully meets all of its design criteria. It is directly applicableto metal blading in large numbers, has frequency response extending fromDC to 100 kHz, does not introduce flow disturbance, and is well suitedto rotating frame applications. These gauges are now being extensivelyused in the turbine research program at MIT.

It will be understood that various changes in details, materials, stepsand arrangement of parts, which have been herein described andillustrated in order to explain the nature of the invention, may be madeby those skilled in the art within the principle and scope of theinvention as expressed in the appended claims.

What is claimed is:
 1. A heat flux multilayered gauge configuration formeasuring a heat flux on an aircraft comprising:a first serpentinepattern resistance temperature sensor; a second serpentine patternresistance temperature sensor; a polyimide substrate locatedintermediate said first resistance temperature sensor and said secondtemperature sensor and bonded thereto, and said substrate furtherlocated on said aircraft blade such that the heat flux to be measuredflows through said substrate, said substrate including a pair ofapertures; a first pair of leading extending from an edge of saidsubstrate to respective ends of said first resistance temperaturesensor; and a second pair of leads extending from an edge of saidsubstrate through respective members of said substrate apertures torespective ends of said second resistance temperature sensor.
 2. A heatflux gauge configuration according to claim 1 wherein said firstserpentine pattern resistance temperature sensor and said secondserpentine pattern resistance temperature sensor are comprised of nickelmaterial and said substrate is comprised of said polyimide.
 3. A heatflux gauge configuration according to claim 1 wherein said firstserpentine pattern resistance temperature sensor and said secondserpentine pattern resistance temperature sensor are comprised ofplatinum material and said substrate is comprised of said polyimide. 4.A heat flux gauge configuration according to claim 1 wherein said firstserpentine pattern resistance temperature sensor and said secondserpentine pattern resistance temperature sensor are comprised oftitanium material and said substrate is comprised of said polyimide. 5.A plurality of heat flux gauge configurations for measuring a heat fluxon an aircraft blade comprising:a plurality of spaced first serpentinepattern resistance temperature sensors; a plurality of spaced secondserpentine pattern resistance temperature sensors; a polyimide substratelocated such that the heat flux to be measured flows through saidsubstrate and said substrate further located intermediate said first andsecond resistance temperature sensors, said substrate including aplurality of pairs of apertures; said plurality of first resistancetemperature sensors, said plurality of second resistance temperaturesensors and said plurality of pairs of apertures all being equal innumber and each having respective members aligned with each other forforming a plurality of heat flux gauge configurations; a plurality offirst pairs of leads extending from an edge of said substrate torespective ends of said first resistance temperature sensors; and aplurality of second pairs of leads extending from an edge of saidsubstrate through respective members of said pair of substrate aperturesin respective ends of said second resistance temperature sensors.
 6. Aplurality of heat flux gauge configurations according to claim 5 whereinsaid plurality of spaced first serpentine pattern resistance temperaturesensors and said plurality of spaced second serpentine patternresistance temperature sensors are each comprised of nickel material andsaid substrate is comprised of said polyimide.